If 3 - 2i is a root of the quadratic equation x^2 + bx + c = 0, where b and c are real num… — ACT

If $3 - 2i$ is a root of the quadratic equation $x^2 + bx + c = 0$ , where $b$ and $c$ are real numbers and $i = \sqrt{-1}$ , what is the value of $b + c$ ?

A
-11
B
-5
C
-1
7Answer
E
19

Answer

The correct answer is

7

. Since the quadratic equation has real coefficients, the roots must be complex conjugates. The conjugate of the root

3 - 2i

3 + 2i

. Vieta's formulas show that the sum of the roots is

-b

, meaning

(3 - 2i) + (3 + 2i) = 6 = -b

, which yields

b = -6

. The product of the roots is

c

, meaning

(3 - 2i)(3 + 2i) = 9 - 4i^2 = 9 - 4(-1) = 13

. Adding these coefficients together gives

b + c = -6 + 13 = 7

Step-by-Step Solution

Identify the second root of the quadratic equation

The second root is

3 + 2i

Since the quadratic equation has real coefficients, the Complex Conjugate Theorem dictates that if a complex number is a root, its complex conjugate must also be a root.

Determine the coefficient

b

using the sum of the roots

b = -6

According to Vieta's formulas, the sum of the roots of the equation

x^2 + bx + c = 0

is equal to

-b

. Thus,

(3 - 2i) + (3 + 2i) = 6 = -b

, which simplifies to

b = -6

Determine the constant term

c

using the product of the roots

c = 13

According to Vieta's formulas, the product of the roots of the equation

x^2 + bx + c = 0

is equal to

c

. Thus,

(3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - 4i^2

. Substituting

i^2 = -1

yields

9 - 4(-1) = 9 + 4 = 13

Calculate the value of

b + c

b + c = 7

Substitute the calculated values of

b

and

c

into the expression:

-6 + 13 = 7

Key Concept

The Complex Conjugate Theorem states that complex roots of polynomials with real coefficients occur in conjugate pairs. Vieta's formulas state that for a quadratic equation

x^2 + bx + c = 0

, the sum of the roots is

-b

and the product of the roots is

c

Question

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Step-by-Step Solution

Key Concept

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